Chapter 8: Infinite Sequences and Series

Section 8.3: Convergence Tests

Example 8.3.19

$$ Determine if the series $\sum _{n\=1}^{\infty }\frac{2\cdot 4\cdot \cdots \cdot \left(2 n\right)}{n\!}$ diverges, converges absolutely, or converges conditionally. If it converges conditionally, determine if it also converge absolutely. $$ Solution $$ Since the numerator of the general term in the series can be written as $2^n\cdot n\!$, this term is actually $a_n\=2^n\cdot n\!\/n\=2^n$. Clearly, $a_n\to \infty$ as $n\to \infty$, so the series diverges by the nth-term test. $$

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